# Questions tagged [calculus-of-variations]

This tag is for problems relating to the calculus of variations that deal with maximizing or minimizing functionals. This problem is a generalization of the problem of finding extrema of functions of several variables. In fact, these variables will themselves be functions and we will be finding extrema of “functions of functions” or functionals.

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### deriving the variation (small change) of the extension of a string [closed]

what is the process to get from the first line (extension of a string) to the second line (small deviation of the extension in the string). Thanks picture
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### Minimizing $(f')^2 + f^2$

I've confused myself when thinking about the following variational problem: \begin{equation} \min_{f} \int_0^T \left([f'(x)]^2 + [f(x)]^2\right)\,dx \qquad f(0) = 1, f'(0) = 0. \tag{*} \end{equation} ...
1 vote
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### Variational Problem.

How to solve variational problem $$I(y)=\int_0^1 [(y’)^2-y|y|y’+xy]dx,y(0)=y(1)=0?$$ I tried by Euler equation, which is $$-2|y|y’+x-\frac{d}{dx}(2y’-y|y|)=0$$ Now stuck. Unable to creat ...
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### How should I use the Jacobi equation to determine the nature of this stationary path? [closed]

Let $n>1$ be a positive integer such that the functional $S[y]=\int_{0}^{1}(y')^{n}e^{y}dx, y(0)=1, y(1)=A>1$ has a stationary path given by $y=n\cdot ln(cx+e^{1/n})$, where $c=e^{A/n}-e^{1/n}$. ...
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### Finding approximation to eigenvalues from minizing Rayleigh quotient and from diagonalization

Suppose I want to find an approximation to the smallest eigenvalue of $y''=\lambda y$, with $y(0)=y(1)=0$. One way to do it is to write the ansatz $y=x(1-x)+ax^2(1-x)^2$, compute the Rayleigh quotient,...
1 vote
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### Dimension of null space of an operator $T$

Let, $K(x,y)$ be a kernal in $[0,1]\times [0,1]$ defined as $K(x,y)=\sin(2\pi x)\sin(2\pi y)$. Consider the integral operator $$T(u)(x)=\int_0^1 u(y)K(x,y)\,dy$$ where, $u\in C[0,1]$. Which of the ...
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### How should I find conditions on the constants $b_{0}, c_{0}, \omega, \gamma$?

According to an economic model, the budget $b(t)$ at time $t\geq 0$ in a household is chosen to maximise the lifetime utility $U[b]=\int_{0}^{\infty}e^{-\beta t}u(c(t))dt$, where $u(c)\geq 0$ is the ...
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### How should I calculate the Euler-Lagrange equation in this problem?

According to an economic model, the budget $b(t)$ at time $t\geq 0$ in a household is chosen to maximise the lifetime utility $U[b]=\int_{0}^{\infty}e^{\beta t}u(c(t))dt$, where $u(c)\geq 0$ is the ...
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1 vote
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### A problem with the calculus of variations

$$\int\limits_{0}^{1}x(t)\sqrt{1+u^2(t)}dt\to inf, \; \dot{x}(t)=u(t), \; u\in PC^1[0,1], \; x(t)\geq 0, \; |u(t)|\leq k, \; x(\pm 1)=R$$ I'm having trouble finding an optimisation. I decided to use ...
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### Finding two coupled Euler-Lagrange equations giving the stationary path of the functional $S[y_{1},y_{2}]=\int[y_{1}'^2+2y_{2}'^2+(2y_{1}+y_{2})^2]dx$

Find the two coupled Euler-Lagrange equations giving the stationary path of the functional $$S[y_{1}, y_{2}]=\int \left[y_{1}'^2+2y_{2}'^2+(2y_{1}+y_{2})^2\right]dx$$ Here's my work: Consider the ...